# Properties

 Label 630.2.bo Level 630 Weight 2 Character orbit bo Rep. character $$\chi_{630}(89,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 32 Newform subspaces 2 Sturm bound 288 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 630.bo (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$105$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$288$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(630, [\chi])$$.

Total New Old
Modular forms 320 32 288
Cusp forms 256 32 224
Eisenstein series 64 0 64

## Trace form

 $$32q - 16q^{4} + O(q^{10})$$ $$32q - 16q^{4} + 12q^{10} - 16q^{16} + 48q^{19} - 12q^{25} + 24q^{31} - 12q^{40} - 16q^{46} - 56q^{49} + 48q^{61} + 32q^{64} - 28q^{70} + 8q^{79} + 128q^{85} - 56q^{91} - 120q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(630, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
630.2.bo.a $$16$$ $$5.031$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-8$$ $$0$$ $$-6$$ $$0$$ $$q-\beta _{8}q^{2}+(-1+\beta _{8})q^{4}-\beta _{7}q^{5}+(-\beta _{10}+\cdots)q^{7}+\cdots$$
630.2.bo.b $$16$$ $$5.031$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$8$$ $$0$$ $$6$$ $$0$$ $$q+(1-\beta _{8})q^{2}-\beta _{8}q^{4}+\beta _{1}q^{5}-\beta _{3}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(630, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(630, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T + T^{2} )^{8}$$)($$( 1 - T + T^{2} )^{8}$$)
$3$ ()()
$5$ ($$1 + 6 T + 21 T^{2} + 54 T^{3} + 113 T^{4} + 168 T^{5} + 186 T^{6} + 84 T^{7} - 6 T^{8} + 420 T^{9} + 4650 T^{10} + 21000 T^{11} + 70625 T^{12} + 168750 T^{13} + 328125 T^{14} + 468750 T^{15} + 390625 T^{16}$$)($$1 - 6 T + 21 T^{2} - 54 T^{3} + 113 T^{4} - 168 T^{5} + 186 T^{6} - 84 T^{7} - 6 T^{8} - 420 T^{9} + 4650 T^{10} - 21000 T^{11} + 70625 T^{12} - 168750 T^{13} + 328125 T^{14} - 468750 T^{15} + 390625 T^{16}$$)
$7$ ($$1 + 14 T^{2} + 173 T^{4} + 1338 T^{6} + 10424 T^{8} + 65562 T^{10} + 415373 T^{12} + 1647086 T^{14} + 5764801 T^{16}$$)($$1 + 14 T^{2} + 173 T^{4} + 1338 T^{6} + 10424 T^{8} + 65562 T^{10} + 415373 T^{12} + 1647086 T^{14} + 5764801 T^{16}$$)
$11$ ($$1 + 38 T^{2} + 930 T^{4} + 16780 T^{6} + 233189 T^{8} + 2491116 T^{10} + 21555910 T^{12} + 152846786 T^{14} + 1257598116 T^{16} + 18494461106 T^{18} + 315600078310 T^{20} + 4413163952076 T^{22} + 49986133101509 T^{24} + 435229984804780 T^{26} + 2918738390350530 T^{28} + 14430493676163158 T^{30} + 45949729863572161 T^{32}$$)($$1 + 38 T^{2} + 930 T^{4} + 16780 T^{6} + 233189 T^{8} + 2491116 T^{10} + 21555910 T^{12} + 152846786 T^{14} + 1257598116 T^{16} + 18494461106 T^{18} + 315600078310 T^{20} + 4413163952076 T^{22} + 49986133101509 T^{24} + 435229984804780 T^{26} + 2918738390350530 T^{28} + 14430493676163158 T^{30} + 45949729863572161 T^{32}$$)
$13$ ($$( 1 + 32 T^{2} + 473 T^{4} + 8616 T^{6} + 148784 T^{8} + 1456104 T^{10} + 13509353 T^{12} + 154457888 T^{14} + 815730721 T^{16} )^{2}$$)($$( 1 + 32 T^{2} + 473 T^{4} + 8616 T^{6} + 148784 T^{8} + 1456104 T^{10} + 13509353 T^{12} + 154457888 T^{14} + 815730721 T^{16} )^{2}$$)
$17$ ($$( 1 + 26 T^{2} + 122 T^{4} + 648 T^{5} + 1104 T^{6} + 39888 T^{7} + 57407 T^{8} + 678096 T^{9} + 319056 T^{10} + 3183624 T^{11} + 10189562 T^{12} + 627576794 T^{14} + 6975757441 T^{16} )^{2}$$)($$( 1 + 26 T^{2} + 122 T^{4} - 648 T^{5} + 1104 T^{6} - 39888 T^{7} + 57407 T^{8} - 678096 T^{9} + 319056 T^{10} - 3183624 T^{11} + 10189562 T^{12} + 627576794 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 - 12 T + 93 T^{2} - 540 T^{3} + 2231 T^{4} - 6864 T^{5} + 8928 T^{6} + 52848 T^{7} - 355902 T^{8} + 1004112 T^{9} + 3223008 T^{10} - 47080176 T^{11} + 290746151 T^{12} - 1337093460 T^{13} + 4375266933 T^{14} - 10726460868 T^{15} + 16983563041 T^{16} )^{2}$$)($$( 1 - 12 T + 93 T^{2} - 540 T^{3} + 2231 T^{4} - 6864 T^{5} + 8928 T^{6} + 52848 T^{7} - 355902 T^{8} + 1004112 T^{9} + 3223008 T^{10} - 47080176 T^{11} + 290746151 T^{12} - 1337093460 T^{13} + 4375266933 T^{14} - 10726460868 T^{15} + 16983563041 T^{16} )^{2}$$)
$23$ ($$( 1 + 4 T - 23 T^{2} + 132 T^{3} + 1031 T^{4} - 2792 T^{5} + 23064 T^{6} + 114968 T^{7} - 503054 T^{8} + 2644264 T^{9} + 12200856 T^{10} - 33970264 T^{11} + 288516071 T^{12} + 849597276 T^{13} - 3404825447 T^{14} + 13619301788 T^{15} + 78310985281 T^{16} )^{2}$$)($$( 1 - 4 T - 23 T^{2} - 132 T^{3} + 1031 T^{4} + 2792 T^{5} + 23064 T^{6} - 114968 T^{7} - 503054 T^{8} - 2644264 T^{9} + 12200856 T^{10} + 33970264 T^{11} + 288516071 T^{12} - 849597276 T^{13} - 3404825447 T^{14} - 13619301788 T^{15} + 78310985281 T^{16} )^{2}$$)
$29$ ($$( 1 - 110 T^{2} + 5617 T^{4} - 185030 T^{6} + 5268820 T^{8} - 155610230 T^{10} + 3972797377 T^{12} - 65430565310 T^{14} + 500246412961 T^{16} )^{2}$$)($$( 1 - 110 T^{2} + 5617 T^{4} - 185030 T^{6} + 5268820 T^{8} - 155610230 T^{10} + 3972797377 T^{12} - 65430565310 T^{14} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 - 6 T + 21 T^{2} - 54 T^{3} - 715 T^{4} + 4356 T^{5} - 5778 T^{6} - 5376 T^{7} + 109626 T^{8} - 166656 T^{9} - 5552658 T^{10} + 129769596 T^{11} - 660317515 T^{12} - 1545974154 T^{13} + 18637577301 T^{14} - 165075684666 T^{15} + 852891037441 T^{16} )^{2}$$)($$( 1 - 6 T + 21 T^{2} - 54 T^{3} - 715 T^{4} + 4356 T^{5} - 5778 T^{6} - 5376 T^{7} + 109626 T^{8} - 166656 T^{9} - 5552658 T^{10} + 129769596 T^{11} - 660317515 T^{12} - 1545974154 T^{13} + 18637577301 T^{14} - 165075684666 T^{15} + 852891037441 T^{16} )^{2}$$)
$37$ ($$1 + 184 T^{2} + 18359 T^{4} + 1197704 T^{6} + 54361073 T^{8} + 1562275568 T^{10} + 12022191570 T^{12} - 1443189401472 T^{14} - 83461258079714 T^{16} - 1975726290615168 T^{18} + 22531522575022770 T^{20} + 4008371682953075312 T^{22} +$$$$19\!\cdots\!33$$$$T^{24} +$$$$57\!\cdots\!96$$$$T^{26} +$$$$12\!\cdots\!79$$$$T^{28} +$$$$16\!\cdots\!76$$$$T^{30} +$$$$12\!\cdots\!41$$$$T^{32}$$)($$1 + 184 T^{2} + 18359 T^{4} + 1197704 T^{6} + 54361073 T^{8} + 1562275568 T^{10} + 12022191570 T^{12} - 1443189401472 T^{14} - 83461258079714 T^{16} - 1975726290615168 T^{18} + 22531522575022770 T^{20} + 4008371682953075312 T^{22} +$$$$19\!\cdots\!33$$$$T^{24} +$$$$57\!\cdots\!96$$$$T^{26} +$$$$12\!\cdots\!79$$$$T^{28} +$$$$16\!\cdots\!76$$$$T^{30} +$$$$12\!\cdots\!41$$$$T^{32}$$)
$41$ ($$( 1 + 84 T^{2} + 9077 T^{4} + 448128 T^{6} + 24944604 T^{8} + 753303168 T^{10} + 25649432597 T^{12} + 399008756244 T^{14} + 7984925229121 T^{16} )^{2}$$)($$( 1 + 84 T^{2} + 9077 T^{4} + 448128 T^{6} + 24944604 T^{8} + 753303168 T^{10} + 25649432597 T^{12} + 399008756244 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 - 196 T^{2} + 20648 T^{4} - 1438284 T^{6} + 72218798 T^{8} - 2659387116 T^{10} + 70591403048 T^{12} - 1238987157604 T^{14} + 11688200277601 T^{16} )^{2}$$)($$( 1 - 196 T^{2} + 20648 T^{4} - 1438284 T^{6} + 72218798 T^{8} - 2659387116 T^{10} + 70591403048 T^{12} - 1238987157604 T^{14} + 11688200277601 T^{16} )^{2}$$)
$47$ ($$( 1 - 30 T + 585 T^{2} - 8550 T^{3} + 104411 T^{4} - 1086708 T^{5} + 9949392 T^{6} - 80614644 T^{7} + 585105414 T^{8} - 3788888268 T^{9} + 21978206928 T^{10} - 112825284684 T^{11} + 509492372891 T^{12} - 1960899809850 T^{13} + 6305840967465 T^{14} - 15198693613890 T^{15} + 23811286661761 T^{16} )^{2}$$)($$( 1 + 30 T + 585 T^{2} + 8550 T^{3} + 104411 T^{4} + 1086708 T^{5} + 9949392 T^{6} + 80614644 T^{7} + 585105414 T^{8} + 3788888268 T^{9} + 21978206928 T^{10} + 112825284684 T^{11} + 509492372891 T^{12} + 1960899809850 T^{13} + 6305840967465 T^{14} + 15198693613890 T^{15} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 - 8 T - 104 T^{2} + 744 T^{3} + 6851 T^{4} - 22388 T^{5} - 596424 T^{6} + 31316 T^{7} + 43664392 T^{8} + 1659748 T^{9} - 1675355016 T^{10} - 3333058276 T^{11} + 54057685331 T^{12} + 311137446792 T^{13} - 2305093557416 T^{14} - 9397689118696 T^{15} + 62259690411361 T^{16} )^{2}$$)($$( 1 + 8 T - 104 T^{2} - 744 T^{3} + 6851 T^{4} + 22388 T^{5} - 596424 T^{6} - 31316 T^{7} + 43664392 T^{8} - 1659748 T^{9} - 1675355016 T^{10} + 3333058276 T^{11} + 54057685331 T^{12} - 311137446792 T^{13} - 2305093557416 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$1 - 426 T^{2} + 99991 T^{4} - 16373094 T^{6} + 2068006933 T^{8} - 212127519060 T^{10} + 18234588368746 T^{12} - 1338906778092288 T^{14} + 84873222726826858 T^{16} - 4660734494539254528 T^{18} +$$$$22\!\cdots\!06$$$$T^{20} -$$$$89\!\cdots\!60$$$$T^{22} +$$$$30\!\cdots\!93$$$$T^{24} -$$$$83\!\cdots\!94$$$$T^{26} +$$$$17\!\cdots\!71$$$$T^{28} -$$$$26\!\cdots\!86$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)($$1 - 426 T^{2} + 99991 T^{4} - 16373094 T^{6} + 2068006933 T^{8} - 212127519060 T^{10} + 18234588368746 T^{12} - 1338906778092288 T^{14} + 84873222726826858 T^{16} - 4660734494539254528 T^{18} +$$$$22\!\cdots\!06$$$$T^{20} -$$$$89\!\cdots\!60$$$$T^{22} +$$$$30\!\cdots\!93$$$$T^{24} -$$$$83\!\cdots\!94$$$$T^{26} +$$$$17\!\cdots\!71$$$$T^{28} -$$$$26\!\cdots\!86$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)
$61$ ($$( 1 - 12 T + 106 T^{2} - 696 T^{3} - 1526 T^{4} + 2700 T^{5} + 111184 T^{6} - 2327412 T^{7} + 40396639 T^{8} - 141972132 T^{9} + 413715664 T^{10} + 612848700 T^{11} - 21128753366 T^{12} - 587839025496 T^{13} + 5461159682266 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16} )^{2}$$)($$( 1 - 12 T + 106 T^{2} - 696 T^{3} - 1526 T^{4} + 2700 T^{5} + 111184 T^{6} - 2327412 T^{7} + 40396639 T^{8} - 141972132 T^{9} + 413715664 T^{10} + 612848700 T^{11} - 21128753366 T^{12} - 587839025496 T^{13} + 5461159682266 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$1 + 352 T^{2} + 62036 T^{4} + 7763648 T^{6} + 806269898 T^{8} + 74285375456 T^{10} + 6186081613776 T^{12} + 467069870756640 T^{14} + 32434315696789747 T^{16} + 2096676649826556960 T^{18} +$$$$12\!\cdots\!96$$$$T^{20} +$$$$67\!\cdots\!64$$$$T^{22} +$$$$32\!\cdots\!18$$$$T^{24} +$$$$14\!\cdots\!52$$$$T^{26} +$$$$50\!\cdots\!96$$$$T^{28} +$$$$12\!\cdots\!08$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$)($$1 + 352 T^{2} + 62036 T^{4} + 7763648 T^{6} + 806269898 T^{8} + 74285375456 T^{10} + 6186081613776 T^{12} + 467069870756640 T^{14} + 32434315696789747 T^{16} + 2096676649826556960 T^{18} +$$$$12\!\cdots\!96$$$$T^{20} +$$$$67\!\cdots\!64$$$$T^{22} +$$$$32\!\cdots\!18$$$$T^{24} +$$$$14\!\cdots\!52$$$$T^{26} +$$$$50\!\cdots\!96$$$$T^{28} +$$$$12\!\cdots\!08$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$)
$71$ ($$( 1 + 52 T^{2} + 10456 T^{4} + 567388 T^{6} + 78505582 T^{8} + 2860202908 T^{10} + 265704536536 T^{12} + 6661214763892 T^{14} + 645753531245761 T^{16} )^{2}$$)($$( 1 + 52 T^{2} + 10456 T^{4} + 567388 T^{6} + 78505582 T^{8} + 2860202908 T^{10} + 265704536536 T^{12} + 6661214763892 T^{14} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$1 - 312 T^{2} + 45700 T^{4} - 4488720 T^{6} + 369040906 T^{8} - 28689322824 T^{10} + 2019175463824 T^{12} - 122728472139336 T^{14} + 7818824571234835 T^{16} - 654020028030521544 T^{18} + 57341031442960733584 T^{20} -$$$$43\!\cdots\!36$$$$T^{22} +$$$$29\!\cdots\!86$$$$T^{24} -$$$$19\!\cdots\!80$$$$T^{26} +$$$$10\!\cdots\!00$$$$T^{28} -$$$$38\!\cdots\!08$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$)($$1 - 312 T^{2} + 45700 T^{4} - 4488720 T^{6} + 369040906 T^{8} - 28689322824 T^{10} + 2019175463824 T^{12} - 122728472139336 T^{14} + 7818824571234835 T^{16} - 654020028030521544 T^{18} + 57341031442960733584 T^{20} -$$$$43\!\cdots\!36$$$$T^{22} +$$$$29\!\cdots\!86$$$$T^{24} -$$$$19\!\cdots\!80$$$$T^{26} +$$$$10\!\cdots\!00$$$$T^{28} -$$$$38\!\cdots\!08$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$)
$79$ ($$( 1 - 2 T - 55 T^{2} + 1622 T^{3} - 4375 T^{4} - 95596 T^{5} + 995034 T^{6} + 1066704 T^{7} - 79120334 T^{8} + 84269616 T^{9} + 6210007194 T^{10} - 47132556244 T^{11} - 170406604375 T^{12} + 4990985479178 T^{13} - 13369810053655 T^{14} - 38407817972318 T^{15} + 1517108809906561 T^{16} )^{2}$$)($$( 1 - 2 T - 55 T^{2} + 1622 T^{3} - 4375 T^{4} - 95596 T^{5} + 995034 T^{6} + 1066704 T^{7} - 79120334 T^{8} + 84269616 T^{9} + 6210007194 T^{10} - 47132556244 T^{11} - 170406604375 T^{12} + 4990985479178 T^{13} - 13369810053655 T^{14} - 38407817972318 T^{15} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 - 250 T^{2} + 32249 T^{4} - 3769146 T^{6} + 372693188 T^{8} - 25965646794 T^{10} + 1530483393929 T^{12} - 81735093342250 T^{14} + 2252292232139041 T^{16} )^{2}$$)($$( 1 - 250 T^{2} + 32249 T^{4} - 3769146 T^{6} + 372693188 T^{8} - 25965646794 T^{10} + 1530483393929 T^{12} - 81735093342250 T^{14} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$1 - 492 T^{2} + 129352 T^{4} - 23026392 T^{6} + 3057469378 T^{8} - 313387087572 T^{10} + 25702504802560 T^{12} - 1837765181416836 T^{14} + 143644724514270163 T^{16} - 14556938002002757956 T^{18} +$$$$16\!\cdots\!60$$$$T^{20} -$$$$15\!\cdots\!92$$$$T^{22} +$$$$12\!\cdots\!18$$$$T^{24} -$$$$71\!\cdots\!92$$$$T^{26} +$$$$31\!\cdots\!92$$$$T^{28} -$$$$96\!\cdots\!72$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$)($$1 - 492 T^{2} + 129352 T^{4} - 23026392 T^{6} + 3057469378 T^{8} - 313387087572 T^{10} + 25702504802560 T^{12} - 1837765181416836 T^{14} + 143644724514270163 T^{16} - 14556938002002757956 T^{18} +$$$$16\!\cdots\!60$$$$T^{20} -$$$$15\!\cdots\!92$$$$T^{22} +$$$$12\!\cdots\!18$$$$T^{24} -$$$$71\!\cdots\!92$$$$T^{26} +$$$$31\!\cdots\!92$$$$T^{28} -$$$$96\!\cdots\!72$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$)
$97$ ($$( 1 + 438 T^{2} + 104849 T^{4} + 16454334 T^{6} + 1868658564 T^{8} + 154818828606 T^{10} + 9282206583569 T^{12} + 364841738158902 T^{14} + 7837433594376961 T^{16} )^{2}$$)($$( 1 + 438 T^{2} + 104849 T^{4} + 16454334 T^{6} + 1868658564 T^{8} + 154818828606 T^{10} + 9282206583569 T^{12} + 364841738158902 T^{14} + 7837433594376961 T^{16} )^{2}$$)